Relativity of Simultaneity

[vuquta]

This thought experiment is modeled after the train/embankment thought experiment of Einstein.

 

There will be two observers, O and O'.

 

The only change is to turn the problem into a 2-D problem instead of 1-D.

So, each observer is centered in a circle, the two circles are of rest radius r and are in relative motion along a common x-axis.

 

Similar to the lightning strikes, the O circle has lasers at every point aimed at the origin O.

 

Now, when O' and O are coincident, the lasers on the circle of O all fire at the same instant.

 

Conclusion:

1) O will see all the laser strikes as simultaneous.

2) O' will not see the laser strikes as simultaneous. [Edit: Laser stike means when O' is struck by a laser]

3) O agrees O' will not see the laser strikes as simultaneous. [Edit: Laser stike means when O' is struck by a laser]

 

Any objections?

Edited January 22, 2010 by vuquta

[swansont]

How about demonstrating that you can solve 1-D relativity problems before tackling anything more complicated?

[vuquta]

How about demonstrating that you can solve 1-D relativity problems before tackling anything more complicated?

 

Well, since we are on the train/embankment, what did you have in mind?

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How about demonstrating that you can solve 1-D relativity problems before tackling anything more complicated?

 

Tell you what.

 

My undergrad degree was in math from a top school.

 

Further, I was summa cum laude graduate.

 

So, I am really not interested in your unfounded and false insults.

 

Did you see any problem with the conclusions above since that is the issue.

[Sisyphus]

Conclusions 2 and 3 are wrong.

 

They both agree that O is hit by all the lasers at once. However, O1 does not agree that all the lasers fired at once.

[swansont]

Tell you what.

 

My undergrad degree was in math from a top school.

 

Further, I was summa cum laude graduate.

 

So, I am really not interested in your unfounded and false insults.

 

Did you see any problem with the conclusions above since that is the issue.

 

Tell you what: you have several posts in other threads that contain elementary math mistakes. Further, you are the one who claimed that saying algebra is internally consistent is "speculation." So I don't particularly care about your credentials. They are not proof against errors.

[vuquta]

Tell you what: you have several posts in other threads that contain elementary math mistakes. Further, you are the one who claimed that saying algebra is internally consistent is "speculation." So I don't particularly care about your credentials. They are not proof against errors.

 

Originally Posted by swansont

And the only way to come to a contradiction when applying algebra is to do the algebra wrong.

 

This is speculation.

 

I am sure you can point out, given the very specific math in my link, where this applies.

 

Why don't we debate that?

 

I was saying here, and it should have been clear,

 

you can use algrebra to find a contradiction if a theory has one.

 

So, of course I assumed we were talking about this concept.

 

I see you thought I was questioning algebra when in fact I was questioning the theory.

 

For example.

 

Theory: There is a greatest integer.

 

Let g be the greatest said integer.

 

But, if g is an integer than g + 1 is an integer by the induction hypothesis.

 

Also, g + 1 > g.

 

Thus, there is no greatest integer.

 

Note I use algebra to run a theory into a contradiction.

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Conclusions 2 and 3 are wrong.

 

They both agree that O is hit by all the lasers at once. However, O1 does not agree that all the lasers fired at once.

 

I agree with your statement.

 

I see my words were unclear above. By laser strikes I mean when O' is struck by the lasers.

 

O' will be struck by each laser at different times based on the x-coordinate.

 

As such, O' will believe they emitted at different times.

 

I will edit the post because it is unclear.

[Sisyphus]

So, wait. To clarify:

 

The lasers are in a circle around O. (In O1's reference frame, it isn't a circle.)

 

In O's reference frame, the lasers fire when O and O1 are coincident. They all hit O at the same time. Most of them miss O1, except for the ones in the direction of their relative velocity. They don't hit O1 at the same time.

 

In O1's reference frame, the lasers all fire at different times, but hit O at the same time. Most of them miss O1, except for the ones in the direction of their relative velocity. They don't hit O1 at the same time.

[vuquta]

So, wait. To clarify:

 

The lasers are in a circle around O. (In O1's reference frame, it isn't a circle.)

 

In O's reference frame, the lasers fire when O and O1 are coincident. They all hit O at the same time. Most of them miss O1, except for the ones in the direction of their relative velocity. They don't hit O1 at the same time.

 

In O1's reference frame, the lasers all fire at different times, but hit O at the same time. Most of them miss O1, except for the ones in the direction of their relative velocity. They don't hit O1 at the same time.

 

I would write exactly this. I am in complete agreement.

 

What do you calculate as the time from when O and O' are coincident on the clock of O' until the laser strikes O'?

 

O will record r/c for all of them.

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I came up with an easier way to do this rather than a circle.

 

Einstein's Train Embankment Thought Experiment.

 

Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A.

http://www.bartleby.com/173/9.html

 

Below is a picture of Einstein's train embankment thought experiment.

A                           M'  ->v                    B
|                           M                          |

Thus, if synchronized clocks had been placed at M' at the front of M' and the rear of M', infinitesimally close, the front clock would record an earlier time than does the back clock for the light from the lightning strikes to hit the clocks. As Einstein said, Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. http://www.bartleby.com/173/9.html

 

 

Now, we add another observer in the frame of M'.

A                       M'  ->v                B                       M'2  ->v 
|                       M                      |  

Place another observer at M'2 in the frame of M' which is the same distance to B that M' is to B and M' is to A. Place a synchronized clock at M'2.

Run the same experiment.

 

Since the rear clock of M' is the same distance from A to M' as is the distance from B to M'2, both observers are in the same frame, and the lightning strikes are simultaneous in the M frame, then M'2 will read the same clock value as does the rear clock of M' for the time when the light hit the clock. Thus, the frame of M' will conclude the time of the front clock of M' will read an earlier time when the light reached the clock than does the clock at M'2. Also the rear clock of M' will read the same value as the clock of M'2.

 

But, the light at B and the observers M' and M'2, with an origin coincident with B when the lightning strikes, are LT equidistant calculated observers and LT concludes M' and M'2 will read the same time for light to strike their clocks.

 

This seems to be a problem.

[timo]

1) O will see all the laser strikes as simultaneous.

2) O' will not see the laser strikes as simultaneous. [Edit: Laser stike means when O' is struck by a laser]

3) O agrees O' will not see the laser strikes as simultaneous. [Edit: Laser stike means when O' is struck by a laser]

 

Any objections?

 

Assuming I understood the setup correctly (I reduced it to 1D as Swansont suggested, but that should not have an impact other than on the nastiness of the calculation):

@1) O will "see" the laser strikes on O as simultaneous (to be precise: by "see as simultaneous" I mean "assign the same time coordinate to both events") . O' will see the laser strikes on O as simultaneous, too.

@2) O will see the laser strikes on O' as non-simultaneous. O' will also see the laser strikes on O' as non-simultaneous.

@3) So in that sense O and O' agree. This is not exactly surprising: A coordinate tuple (let's just call it a vector for simplicity) [math]p=(t, \vec x)[/math] describing an event (like time and position when a laser is emitted or hits something) transforms with an invertable matrix L from one coordinate system to another: p' = Lp. So for two events with the coordinates [math]p_1[/math] and [math]p_2[/math], say the impacts of two of the lasers, the relations [math]p_1 = p_2 \Rightarrow p_1' = Lp_1 = Lp_2 = p_2'[/math] and [math]p_1 \neq p_2 \Rightarrow p_1' = Lp_1 \neq Lp_2 = p_2'[/math] hold. Note that equality of coordinates means equality of space and time coordinates.

 

I guess I have no objections.

[vuquta]

Assuming I understood the setup correctly (I reduced it to 1D as Swansont suggested, but that should not have an impact other than on the nastiness of the calculation):

@1) O will "see" the laser strikes on O as simultaneous (to be precise: by "see as simultaneous" I mean "assign the same time coordinate to both events") . O' will see the laser strikes on O as simultaneous, too.

@2) O will see the laser strikes on O' as non-simultaneous. O' will also see the laser strikes on O' as non-simultaneous.

@3) So in that sense O and O' agree. This is not exactly surprising: A coordinate tuple (let's just call it a vector for simplicity) [math]p=(t, \vec x)[/math] describing an event (like time and position when a laser is emitted or hits something) transforms with an invertable matrix L from one coordinate system to another: p' = Lp. So for two events with the coordinates [math]p_1[/math] and [math]p_2[/math], say the impacts of two of the lasers, the relations [math]p_1 = p_2 \Rightarrow p_1' = Lp_1 = Lp_2 = p_2'[/math] and [math]p_1 \neq p_2 \Rightarrow p_1' = Lp_1 \neq Lp_2 = p_2'[/math] hold. Note that equality of coordinates means equality of space and time coordinates.

 

I guess I have no objections.

 

Yes, I agree wth your assessment. Thanks for your time.

 

Oh, I added an additional feature in a prior post.

 

I say we remain 1-D.

 

An additional observer is added.

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Assuming I understood the setup correctly (I reduced it to 1D as Swansont suggested, but that should not have an impact other than on the nastiness of the calculation):

@1) O will "see" the laser strikes on O as simultaneous (to be precise: by "see as simultaneous" I mean "assign the same time coordinate to both events") . O' will see the laser strikes on O as simultaneous, too.

@2) O will see the laser strikes on O' as non-simultaneous. O' will also see the laser strikes on O' as non-simultaneous.

@3) So in that sense O and O' agree. This is not exactly surprising: A coordinate tuple (let's just call it a vector for simplicity) [math]p=(t, \vec x)[/math] describing an event (like time and position when a laser is emitted or hits something) transforms with an invertable matrix L from one coordinate system to another: p' = Lp. So for two events with the coordinates [math]p_1[/math] and [math]p_2[/math], say the impacts of two of the lasers, the relations [math]p_1 = p_2 \Rightarrow p_1' = Lp_1 = Lp_2 = p_2'[/math] and [math]p_1 \neq p_2 \Rightarrow p_1' = Lp_1 \neq Lp_2 = p_2'[/math] hold. Note that equality of coordinates means equality of space and time coordinates.

 

I guess I have no objections.

 

 

I was just looking here.

 

What is your matrix operation so I can see the determinant. Since we are operating in 1-D space, I assume you are using one for time and the x-axis for said matrix.

 

I did not see the need for a bijective nature and thus the need for an inverse matrix.

 

Perhaps you could explain why an inverse matrix is necessary.

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Assuming I understood the setup correctly (I reduced it to 1D as Swansont suggested, but that should not have an impact other than on the nastiness of the calculation):

@1) O will "see" the laser strikes on O as simultaneous (to be precise: by "see as simultaneous" I mean "assign the same time coordinate to both events") . O' will see the laser strikes on O as simultaneous, too.

@2) O will see the laser strikes on O' as non-simultaneous. O' will also see the laser strikes on O' as non-simultaneous.

@3) So in that sense O and O' agree. This is not exactly surprising: A coordinate tuple (let's just call it a vector for simplicity) [math]p=(t, \vec x)[/math] describing an event (like time and position when a laser is emitted or hits something) transforms with an invertable matrix L from one coordinate system to another: p' = Lp. So for two events with the coordinates [math]p_1[/math] and [math]p_2[/math], say the impacts of two of the lasers, the relations [math]p_1 = p_2 \Rightarrow p_1' = Lp_1 = Lp_2 = p_2'[/math] and [math]p_1 \neq p_2 \Rightarrow p_1' = Lp_1 \neq Lp_2 = p_2'[/math] hold. Note that equality of coordinates means equality of space and time coordinates.

 

I guess I have no objections.

 

 

Oh, I just thought of something else.

 

What is your calculation for time in the moving frame from the light emission point to the strike at the moving observer for the positive x-axis laser?

 

This matrix would resolve this matter.

[timo]

What is your matrix operation so I can see the determinant. Since we are operating in 1-D space, I assume you are using one for time and the x-axis for said matrix.

The matrix is the Lorentz transformation [math]L = \left( \begin{array}{cc} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{array} \right)[/math] with beta being the velocity of the moving frame in units of the speed of light (i.e. beta=0 is non-moving, beta=0.5 is moving with half the speed of light and beta=1 is light-speed) and [math]\gamma = \frac{1}{\sqrt{1-\beta ^2}}[/math] the Gamov factor (but the name is irrelevant for calculations).

 

I did not see the need for a bijective nature and thus the need for an inverse matrix. Perhaps you could explain why an inverse matrix is necessary.

And example where a non-invertible matrix screws up my statement: [math] \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \neq \left( \begin{array}{c} 2 \\ 0 \end{array} \right)[/math] but [math] \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)\left( \begin{array}{c} 1 \\ 0 \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} 2 \\ 0 \end{array} \right)[/math]

 

What is your calculation for time in the moving frame from the light emission point to the strike at the moving observer for the positive x-axis laser?

In the frame of O, O' moves on a wold-line given by [math]P_2(x) = (x, \beta x)[/math]. The laser moves on a world-line given by [math]L_+(x) = (x, 1-x)[/math]. They do meet at [math]I = \left(\frac{1}{\beta +1 }, \frac{\beta}{\beta+1} \right)[/math]. In the coordinates of the moving frame, [math] I' = LI = \dots = \frac{1}{1+\beta} \left( \frac 1\gamma, 0 \right)[/math]. I.e. the time the moving observer measures between passing through the origin (time coordinate equals zero) and hitting the light of the laser is [math] \frac{1}{\gamma(1+\beta)}[/math].

[vuquta]

The matrix is the Lorentz transformation [math]L = \left( \begin{array}{cc} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{array} \right)[/math] with beta being the velocity of the moving frame in units of the speed of light (i.e. beta=0 is non-moving, beta=0.5 is moving with half the speed of light and beta=1 is light-speed) and [math]\gamma = \frac{1}{\sqrt{1-\beta ^2}}[/math] the Gamov factor (but the name is irrelevant for calculations).

 

 

And example where a non-invertible matrix screws up my statement: [math] \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \neq \left( \begin{array}{c} 2 \\ 0 \end{array} \right)[/math] but [math] \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)\left( \begin{array}{c} 1 \\ 0 \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} 2 \\ 0 \end{array} \right)[/math]

 

 

In the frame of O, O' moves on a wold-line given by [math]P_2(x) = (x, \beta x)[/math]. The laser moves on a world-line given by [math]L_+(x) = (x, 1-x)[/math]. They do meet at [math]I = \left(\frac{1}{\beta +1 }, \frac{\beta}{\beta+1} \right)[/math]. In the coordinates of the moving frame, [math] I' = LI = \dots = \frac{1}{1+\beta} \left( \frac 1\gamma, 0 \right)[/math]. I.e. the time the moving observer measures between passing through the origin (time coordinate equals zero) and hitting the light of the laser is [math] \frac{1}{\gamma(1+\beta)}[/math].

 

Let's see, I am going to ignore your inverse concepts because you did not explain why an invertible matrix was necessary. You brought it up.

 

I guess that is because you do not know why.

 

But, I will focus on a more astounding concept.

 

[math] \frac{1}{\gamma(1+\beta)}[/math]

 

This was your timing concept for the light ray.

 

Where is your distance?

[mooeypoo]

vuquta, get yourself off your high horse. If you don't know the subject, the least you can do is go over its basics, and if you can't do that, the least you could do then is *not* blame others for having faulty math when you are the one who doesn't know it properly.

 

As it turns out, even wikipedia can be an answer here: http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form

 

Read. Absorb. Learn. Stop whining about other people's math when yours is the one at fault here.

 

~moo

[vuquta]

vuquta, get yourself off your high horse. If you don't know the subject, the least you can do is go over its basics, and if you can't do that, the least you could do then is *not* blame others for having faulty math when you are the one who doesn't know it properly.

 

As it turns out, even wikipedia can be an answer here: http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form

 

Read. Absorb. Learn. Stop whining about other people's math when yours is the one at fault here.

 

~moo

 

 

Yea, is there any way you can supply math to support your assertions?

 

That is what is going on in this thread.

 

Now, I would assume you can go through my statements in a specific way and refute them mathematically.

 

I assume you are able to handle mathematical debate, correct?

 

So, let's go.

[mooeypoo]

Yea, is there any way you can supply math to support your assertions?

I did.. look at the link.

 

You said this:

Let's see, I am going to ignore your inverse concepts because you did not explain why an invertible matrix was necessary. You brought it up.

 

I guess that is because you do not know why.

When in fact, you are the one who doesn't know why. I helped out by giving you the link to show why.

You're welcome.

 

You claimed that you are going to ignore the matrix because you don't know what it means. I gave you the page to explain to you what it means. Go read it.

 

~moo

[vuquta]

vuquta, get yourself off your high horse. If you don't know the subject, the least you can do is go over its basics, and if you can't do that, the least you could do then is *not* blame others for having faulty math when you are the one who doesn't know it properly.

 

As it turns out, even wikipedia can be an answer here: http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form

 

Read. Absorb. Learn. Stop whining about other people's math when yours is the one at fault here.

 

~moo

 

 

OK, so answer this,

 

 

 

But, I will focus on a more astounding concept.

 

[math] \frac{1}{\gamma(1+\beta)}[/math]

 

This was your timing concept for the light ray.

 

Where is your distance?

 

The poster did not answer. You answer.

[mooeypoo]

This is the Lorentz transformation factor, vuquta. It's the transition factor between reference frames. You use this to find the distance or time, relative to different frames.

 

Have you even studied Special Relativity? If you have, did you not derive this?

[vuquta]

I did.. look at the link.

 

You said this:

 

When in fact, you are the one who doesn't know why. I helped out by giving you the link to show why.

You're welcome.

 

You claimed that you are going to ignore the matrix because you don't know what it means. I gave you the page to explain to you what it means. Go read it.

 

~moo

 

BTW, would you like to debate the math of this?

I would.

[mooeypoo]

BTW, would you like to debate the math of this?

I would.

I'm sure.

 

You didn't answer my question -- did you derive this equation, ever?

[vuquta]

This is the Lorentz transformation factor, vuquta. It's the transition factor between reference frames. You use this to find the distance or time, relative to different frames.

 

Have you even studied Special Relativity? If you have, did you not derive this?

 

 

Well, it is missing a distance term.

 

If you have studied this, you would have noticed.

 

Why did you miss this?

 

So, why, since you know everything going on, why dont you tell me when O' will record the light?

 

Then, since I am stupid, we will go from there.

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I'm sure.

 

You didn't answer my question -- did you derive this equation, ever?

 

Answer, Yes, and this is different fron deriving LT. Did you know that?

 

I can do both.

[mooeypoo]

Well, it is missing a distance term.

Nope, it doesn't. Derive it.

 

If you have studied this, you would have noticed.

I did. I derived it. Did you? You should.

 

The derivation itself shows you not only what the equation means, but where it came from. Good exercise, too. Not too hard.

 

Why did you miss this?

Okay, listen. what is velocity?

 

v = dx/dt

 

You *have* a treatment for 'distance' in there. In the form of a differential - the velocity. The formula doesn't ignore it. If you had derived it like you should do, you'd see it.

 

Seriously, vuquta, your refusal to listen serves no good to the debate or to your own misconceptions. It's quite obvious your math is faulty, if you're so sure you're right, you should at least go over the way the original formula you're protesting was derived. Maybe you'll learn something, or at least show us better where it's wrong.

 

As it is atm, you're making no sense.

 

So, why, since you know everything going on, why dont you tell me when O' will record the light?

I'm not getting into your scenario when your fundamentals are wrong.

 

Then, since I am stupid, we will go from there.

I didn't say you're stupid, but your refusal to cooperate is not just annoying, it's counter productive.

 

You don't know the math. That doesn't make you stupid, it just makes you not-knowledgeable of the math. Or confused. Either way, we don't need to guess it, you're showing it.

 

 

Answer, Yes, and this is different fron deriving LT. Did you know that?

 

I can do both.

Go ahead, then.

[vuquta]

OK,

 

With all these timid humans here.

 

Here is is.

 

let r be the distance to B

 

t = r/(λ(c+v))

 

http://www.tc.umn.edu/~janss011/pdf%20files/appendix-SR.pdf

Page 17.

[mooeypoo]

Page 17 already uses the Lorentz transform. What I meant is for you to derive the lorentz transform.

 

You're the one claiming it is missing. I'm asking you to derive it so you show me which of the steps that are used to derive it is the faulty one.

 

I'm waiting.

 

 

And.. uhm... if it wasn't clear, I mean a MATHEMATICAL derivation. Not a word-explanation like pg 17 has, which is lacking mathematically.

[vuquta]

Nope, it doesn't. Derive it.

 

 

I did. I derived it. Did you? You should.

 

The derivation itself shows you not only what the equation means, but where it came from. Good exercise, too. Not too hard.

 

 

Okay, listen. what is velocity?

 

v = dx/dt

 

You *have* a treatment for 'distance' in there. In the form of a differential - the velocity. The formula doesn't ignore it. If you had derived it like you should do, you'd see it.

 

Seriously, vuquta, your refusal to listen serves no good to the debate or to your own misconceptions. It's quite obvious your math is faulty, if you're so sure you're right, you should at least go over the way the original formula you're protesting was derived. Maybe you'll learn something, or at least show us better where it's wrong.

 

As it is atm, you're making no sense.

 

 

I'm not getting into your scenario when your fundamentals are wrong.

 

 

I didn't say you're stupid, but your refusal to cooperate is not just annoying, it's counter productive.

 

You don't know the math. That doesn't make you stupid, it just makes you not-knowledgeable of the math. Or confused. Either way, we don't need to guess it, you're showing it.

 

 

 

Go ahead, then.

 

I am not going through all this.

 

You are off task of this thread.

 

If I had game, I would simply do the math and refute.

You do not and cannot.

So, let's get back on task of this thead.

[mooeypoo]

I am not going through all this.

Why not?

 

You claimed the transformation is faulty. Your explanation as to why is lacking. Either show us why it's faulty or go over the derivation to see that it isn't.

 

You are off task of this thread.

Not when your claims are lacking in the basic principles. In that case, we must go over the basic principles first before we move on to the more complicated case.

 

If I had game, I would simply do the math and refute.

I don't know what "If I had game" means, but you are the one making the claim, and you seem to be unable to prove yourself.

 

Either stop saying the math is faulty and start using the PROPER math, or show us where it's faulty using the math.

 

You can't eat the cake and leave it whole, vuquta. You can't just decide you're using your own set of mathematical formulas because the original ones are flawed but not know to explain (and *prove*) where they're flawed.

 

We can't go on without it, not when you insist on not using proper math.

 

 

 

You do not and cannot.

I'm not the one making the claim that I know better than what the theory states.

 

So, let's get back on task of this thead.

Only if you agree to use proper math. Which you don't. So if you don't, you need to first prove your math concepts otherwise we can't move on.

 

Really, it's that simple.

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Here, a good resource for you to go over: http://www.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf --> a derivation (with explanation) of Lorentz transform, with an explanation about the nature of inertial frames.

 

~moo